Question

[10 points]

find the range of each of the following function. All the function are definied for all values of x.

f(x) = 3(x+5)² +2

Can you give the answer and explanation?
what does 'all the function are definied for all values of x'?

Heldpdlsoemss​

Answer 1

Hey there!

First, we must understand what range is. Range is just another word for all the possible y values of the equation.

This equation, f(x) = 3(x+5)² +2 is in vertex form. We see that this creates a parabola with a vertex at (-5, 2) and opens up.

Thus, the range of this graph would be: [2,∞) in interval notation. Note, the [ is used for greater than or equal to, or less than or equal to, and the ")" is used for less than or greater than.

We use ) for ∞, because ∞ is just a concept, not a real number.

Finally, all functions are defined for all values of x means that this is an f(x) function, and the f(x) is equal to something that can be written in terms of x.

Let me know if this helps :)

Answer 2

y ≥ 2

Explanation:

The range is all of the values that f(x) can have. In order to find the range, let's first find the vertex. That's pretty easy to do since f(x) is already written in vertex notation (vertex notation is f(x) = a(x - c)² + d where (c, d) is the vertex). Since c = -5 and d = 2, the vertex is (-5, 2). Because the value of a is positive (a = 3 in this case, which is positive) we know that the parabola opens upward, therefore the vertex is the minimum. This means that the y-coordinate of the vertex is the smallest possible y-value of the function, therefore, since the y-coordinate of the vertex is 2, the range is y ≥ 2. Note that we use ≥ and not >, this is because y can be 2, therefore, it is included in the solution set.